Decompressing Apparatus and Compressing Apparatus

1. A decompressing apparatus comprising: an input microprocessor that inputs trace expression data, in which an element in a subgroup of a multiplicative group of a finite field is trace-expressed, and additional data, the additional data being obtained based on the trace expression data and affine expression data in which the trace expression data is affine-expressed; a calculating microprocessor that calculates a plurality of solutions of simultaneous equations derived by the trace expression data; a first selecting microprocessor that selects any of a plurality of items of affine expression data in which the element is affine-expressed based on the additional data, the affine expression data being found from the solutions; and a decompressing microprocessor that decompresses the selected affine expression data to the element, wherein the calculating microprocessor comprises: a first solution finding microprocessor that finds solutions of a first equation obtained by inputting the trace expression data into a preset coefficient of a previously-found k 1th -degree equation (k 1 is a preset integer of 1 or more) over the finite field; and a second solution finding microprocessor that finds solutions of a second equation obtained by inputting at least one of the solutions of the first equation into a preset coefficient of a previously-found k 2th -degree equation (k 2 is a preset integer of 1 or more) over the finite field, and the calculating microprocessor calculates a plurality of solutions of the simultaneous equations based on the solutions of the second equation.

2. The apparatus according to claim 1 , wherein the first solution finding microprocessor finds two solutions of the first equation obtained by inputting the trace expression data into a preset coefficient in a previously-found quadric equation over the finite field, the second solution finding microprocessor finds three solutions of the second equation obtained by inputting at least one of the two solutions into a preset coefficient of a previously-found cubic equation over the finite field, the calculating microprocessor further comprises a third solution finding unit that finds solutions of a third equation obtained by inputting at least one of the two solutions of the first equation and the three solutions of the second equation into a preset coefficient of a previously-found linear equation over the finite field, and the calculating microprocessor calculates a plurality of solutions of the simultaneous equations based on the solutions of the second equation and the solutions of the third equation.

3. The apparatus according to claim 1 , wherein the calculating microprocessor comprises a second selecting microprocessor that selects any of the solutions of the first equation based on the additional data, and the second solution finding microprocessor finds solutions of the second equation obtained by inputting a solution selected by the second selecting microprocessor into a preset coefficient of the k 2th -degree equation.

4. A compressing apparatus comprising: a first transforming microprocessor that transforms an element in a subgroup of a multiplicative group of a finite field into trace expression data in which the element is trace-expressed; a second transforming microprocessor that transforms the element into affine expression data in which the element is affine-expressed; a deciding microprocessor that decides additional data for finding the affine expression data based on the affine expression data; an output microprocessor that outputs the trace expression data and the additional data; and a calculating microprocessor that finds solutions of a first equation obtained by inputting the trace expression data into a preset coefficient of a previously-found k 1th -degree equation (k 1 is a preset integer of 1 or more) over the finite field, finds solutions of a second equation obtained by inputting at least one of the solutions of the first equation into a preset coefficient of a previously-found k 2th -degree equation (k 2 is a preset integer of 1 or more) over the finite field, and calculates a plurality of solutions of the simultaneous equations based on the solutions of the second equation, wherein the deciding microprocessor decides, as the additional data, the order of a solution coincident with the sum of elements of the affine expression data from a head when the solutions of the first equation are arranged in a preset rule, and the order of a solution coincident with the affine expression data from the head when the solutions of the simultaneous equations are arranged in the rule.

5. A program product having a non-transitory computer readable medium including programmed instructions, wherein the instructions, when executed by a computer, cause the computer to perform: inputting trace expression data, the trace expression data including an element in a subgroup of a multiplicative group of a finite field that is trace-expressed, and additional data, the additional data being obtained based on the trace expression data and affine expression data in which the trace expression data is affine-expressed; calculating a plurality of solutions of simultaneous equations derived by the trace expression data; selecting any of a plurality of items of affine expression data in which the element is affine-expressed based on the additional data, the affine expression data being found from the solutions; and decompressing the selected affine expression data to the element, wherein the calculating includes finding solutions of a first equation obtained by inputting the trace expression data into a preset coefficient of a previously-found k 1th -degree equation (k 1 is a preset integer of 1 or more) over the finite field; and finding solutions of a second equation obtained by inputting at least one of the solutions of the first equation into a preset coefficient of a previously-found k 2th -degree equation (k 2 is a preset integer of 1 or more) over the finite field, and calculating a plurality of solutions of the simultaneous equations based on the solutions of the second equation.